Space of subspheres and conformal invariants of curves

Rémi Langevin, 1Jun O’Hara and Shigehiro Sakata 2

Abstract

A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations.
We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the normal form of the curve.
We also give characterization of canal surfaces in terms of curves in the set of circles.

A space curve is determined by the arc-length, curvature, and torsion up to motions of R3.
In a Möbius geometric framework, it was shown by Fialkow ([F]) that a space curve is determined by three conformal invariants, conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations.

The conformal arc-length was found by Liebmann [L] for space curves and by Pick for plane curves.
The conformal curvature and torsion were given by Vessiot [V].
They have been studied by Fialkow using conformal derivations [F], by Sulanke using Cartan’s group theoretical method of moving frames [Su], and by Cairns, Sharpe, and Webb using normal forms [CSW].
Conformal torsion was also studied in [MRS] using conformal invariants for pairs of spheres.

In [LO2] the first and the second authors showed that the conformal arc-length can be considered as “12-dimensional measure” of the curve of osculating circles.
This paper is a natural continuation of it. We use both curves γ of osculating circles and σ of osculating spheres, which belong to different spaces, and by doing so we give (hopefully) new formula for the conformal curvature (theorem 4.6).
We also give moving frames in the Minkowski space using γ and σ.
Then the conformal curvature and conformal torsion appear in Frenet matrix (this is a just of translation of Sulanke’s result [Su] to our context).
By taking the projection to the Euclidean space which can be realized in R51 as an affine section of the light cone, we obtain the normal form given in [CSW].
We give a proof of formulae of conformal curvature and torsion in [CSW] using our theorem 4.6.

Thus, we get geometric and simpler description of integration of preceding studies of conformal invariants of space (and planar) curves.

We also study canal surfaces and give the characterization of them as curves in the set of circles in S3.

Let us start with basics in Möbius geometry which are needed for the study of curves of osculating circles and of osculating spheres.

2.1 Realization of R3 and \boldmathS3 in Minkowski space R51

We start by recalling a commonly used models of a sphere and the Euclidean space in Möbius geometry (cf. [B, Ce, HJ]). Let us explain in a 3-dimensional case.

The Minkowski spaceR51 is R5 endowed with an indefinite inner product (the Minkowski product or the Lorentz form) given by

⟨x,y⟩=−x0y0+x1y1+x2y2+x3y3+x4y4.

The light coneC is given by C={x∈R51|⟨x,x⟩=0}.
A vector subspace W≠RR51 of R51 is said to be time-like if it contains a non-zero time-like vector.
When dimW≥2, W is time-like if and only if it intersects the light cone transversally.

A 3-sphere S3 or R3∪{∞} can be identified with the projectivization of the light cone.
In fact, they can be isometrically embedded in R51 as the intersection of the light cone and a codimension 1 affine subspace H.
When H can be expressed as H={x|⟨x,n⟩=−1} for some unit time-like vector n, in which case H is tangent to the hyperboloid {y|⟨y,y⟩=−1} at point n, H∩C is a unit sphere.
When H can be expressed as H={x|⟨x,n⟩=−1} for some light-like vector n, in which case H is parallel to a codimension 1 subspace which is tangent to the light cone in span(n), H∩C is a paraboloid E3. The metric induced from the Lorentz quadratic form on E3 is Euclidean.
For example, when n=(1/√2,1/√2,0,0,0), the Euclidean space can be isometrically embedded as

We use the notation S3 and E3 to emphasize that they are embedded in R51.

2.2 de Sitter space as the set of codimension 1 spheres

An oriented sphere Σ in S3 can be obtained as the intersection of an oriented time-like 4-dimensional subspace of R51.
Therefore the set S(2,3) of oriented spheres in S3 can be identified with the Grassmann manifold ˜G−4,5 of oriented time-like 4 dimensional subspaces of R51.
By taking the orthogonal complement, we obtain a bijection between ˜G−4,5 and the set of oriented space-like lines, which can be identified with the quadric Λ4={x∈R51|⟨x,x⟩=1} called de Sitter space (Figure 1).
The bijection from Λ4 to S(2,3) is given by

Λ4∋σ↦Σ=S3∩(span(σ))⊥∈S(2,3),

oriented as the boundary of the ball S3∩{⟨σ,.⟩≤0}⊂R51,
and its inverse is given by

S(2,3)∋S3∩span(u1,u2,u3,u4)↦u1×u2×u3×u4∥u1×u2×u3×u4∥∈Λ4,

where w=u1×u2×u3×u4 is the Lorentz vector product with respect to ⟨⋅,⋅⟩ that is characterized by ⟨w,ui⟩=0, the norm ∥w∥=√|⟨w,w⟩| being equal to the volume of parallelepiped spanned by ui which is given by √|det(⟨ui,uj⟩)|, and det(w,u1,u2,u3,u4)>0 ([LO]3). Then w is given by w0=−det(e0,u1,u2,u3,u4) and wi=det(ei,u1,u2,u3,u4)(i≠0). Direct computation shows (W≠R51)

Figure 1: The correspondence between de Sitter space and the set of oriented 2-spheres.

Similarly, the set of oriented circles in S2 can be identified with de Sitter space Λ3 in R41.

2.3 Pseudo-Riemannian structure of indefinite Grassmann manifolds

In general, the set of oriented k-dimensional subspheres in Sn can be identified with the Grassmann manifold ˜G−k+2,n+2 of oriented time-like (k+2)-dimensional subspaces in Rn+21.
By taking the orthogonal complement we obtain a bijection from ˜G−k+2,n+2 to the Grassmann manifold ˜G+n−k,n+2 of oriented space-like (n−k)-dimensional subspaces in Rn+21.
Thus we need the pseudo-Riemannian structure of these Grassmannians.

The canonical extension of positive definite inner product to Grassmann algebras is given by

⟨u1∧⋯∧uq,v1∧⋯∧vq⟩=det(⟨ui,vj⟩).

But in our case, as we start with the Minkowski space, it does not fit (2.1). Therefore we agree (after [HJ] page 280) that the pseudo-Riemannian structures on our Grassmannians are given by

2.4 Two Grassmannians as the set of oriented circles in S3

An oriented circle Γ in S3 can be obtained as the intersection of an oriented time-like 3-dimensional subspace of R51.
Therefore the set S(1,3) of oriented circles in S3 can be identified with the Grassmann manifold ˜G−3,5 of oriented time-like 3 dimensional subspaces of R51.
By taking the orthogonal complement, we obtain the bijection between ˜G−3,5 and the Grassmann manifold ˜G+2,5 of oriented space-like 2 dimensional subspaces of R51.

Recall that the wedge product u∧v of two vectors in R51, u=(u0,u1,…,u4) and v=(v0,v1,…,v4), is given by u∧v=(pij)0≤i<j≤4∈\lx@stackrel2⋀R51≅R10, where the Plücker coordinates pij are given by

pij=∣∣∣uiujvivj∣∣∣,

(2.3)

and a vectore \boldmathp=(pij)0≤i<j≤4 in \lx@stackrel2⋀R51≅R10 is a pure 2-vector, i.e. the wedge product of two vectors in R51 if and only if pij satisfy the Plücker relation:

p01p23−p02p13+p03p12

=

0,

(2.4)

p01p24−p02p14+p04p12

=

0,

(2.5)

p01p34−p03p14+p04p13

=

0,

(2.6)

p02p34−p03p24+p04p23

=

0,

(2.7)

p12p34−p13p24+p14p23

=

0.

(2.8)

We remark that all of them are not independent.
For example, the relations (2.7) and (2.8) can be derived from the rest if p01≠0.

Then ˜G+2,5 can be identified with the set of unit space-like pure 2-vectors:

Now the identification between S(1,3) and ˜G+2,5 can be explicitly given by

S(1,3)∋Γ=S3∩(span(u,v))⊥↦u∧v∥u∧v∥∈˜G+2,5.

(2.9)

2.5 How to express osculating circles and osculating spheres

Let m=m(s) be a point in a curve C in E3⊂R51, where s is the arc-length parameter.
We always assume that the differential of m never vanishes in what follows.
Let Π be a time-like vector subspace of R51 of dimension 3 (or 4).
Then the curve C has contact of order ≥k with the circle (or the sphere respectively) Π∩E3 if and only if m(s),m′(s),⋯,m(k)(s) belong to Π (remark that the curve is not in R3 but in E3).
Therefore, an osculating circle to C at m(s) is given by E3∩span(m(s),m′(s),m′′(s)), and an osculating sphere is generically given by E3∩span(m(s),m′(s),m′′(s),m′′′(s)).

In what follows we assume that the curve C is vertex free, i.e. the osculating circles have contact of order exactly equal to 2.
Then the point σ(s) in Λ4 that corresponds to the osculating sphere at point m(s) is given by

σ(s)=m(s)×m′(s)×m′′(s)×m′′′(s)∥m(s)×m′(s)×m′′(s)×m′′′(s)∥.

(2.10)

Note that σ′(s) can be expressed as

σ′(s)=m(s)×m′(s)×m′′(s)×(am′′′(s)+bm(4)(s))

(2.11)

for some a,b∈R.
Let us further assume that ∥σ′(s)∥ never vanishes.
As ⟨σ,σ′⟩=0 we have dimspan(σ,σ′)=2.
The formulae (2.10) and (2.11) imply

span(σ(s),σ′(s))=(span(m(s),m′(s),m′′(s)))⊥.

Therefore, the osculating circle Γ(s) to C at point m(s) corresponds to a point γ(s) in ˜G+2,5 given by

γ(s)=±σ(s)∧σ′(s)∥σ(s)∧σ′(s)∥.

For simplicity’s sake, we assume that the sign is + in what follows.
If we denote the derivative by the arc-length parameter l of the curve of osculating spheres σ in Λ4 by putting \Huge.σ above, we have

γ=σ∧\Huge.σ

(2.12)

An osculating circle to a curve C in E2 (in this case, the osculating sphere is constantly equal to E2) can be expressed in a similar way as (2.10).
In this case, as ∥m(s)×m′(s)×m′′(s)∥=1 ([LO2]), a point γ in Λ3 which corresponds to the osculating circle is given by γ=m(s)×m′(s)×m′′(s).

Similarly, an osculating circle to a curve in E3 can be expressed by

γ(s)=m(s)∧m′(s)∧m′′(s)∈˜G−3,5.

(2.13)

2.6 Our notations and assumptions

We express a curve in E2 or E3 by C and a point on it by m.
We always assume that C is vertex-free.
The letters γ and σ express the osculating circle and sphere respectively unless otherwise mentioned.
There are one or two natural conformally invariant parameters: one is the conformal arc-length which will be denoted by ρ, and the other is the arc-length parameter of the curve of osculating spheres which will be denoted by l.
The latter appears only when C is a space curve.
We also use the arc-length s of C.
In order to avoid confusion, we use different notations to express the derivatives by these parameters.
The derivatives by s,ρ, and l are expressed by putting ′,\tiny∘σ, and \Huge.σ respectively.
For simplicity’s sake, we put a technical assumption that dldρ never vanishes when C is a space curve.

2.7 Null curve of osculating circles and conformal arc-length

Let us first consider a smooth one parameter family of circles in R2 given by their centers ω(t) and radii r(t).
Locally speaking, these circles may admit an envelope formed of two curves, or exceptionally one curve; this is the case we will consider here, because we consider the family of osculating circles to a curve.

Figure 2: Osculating circles of a plane curve with monotone
curvature

Then ∥ω′(t)∥=|r′(t)| everywhere.
The fact that the osculating circles of a planar arc with monotone curvature are nested along the arc as is illustrated in Figure 2 was observed in the beginning of 20th century by Kneser ([K]).

In our language,

Lemma 2.1

([LO2])
A curve γ⊂Λ3 of osculating circles to a curve C in E2 is light-like, and the point m(s) on C can be given by m(s)=E2∩span(γ′(s)).

Similarly, when C is a curve in E3, a curve γ of osculating circles is a null curve in ˜G+2,5 with γ′ being a pure vector.
If Π(s) is a plane in R51 which corresponds to γ′(s)∈\lx@stackrel2⋀R51, then it is tangent to the light cone in a line span(m(s)).

Proposition 2.2

([LO2])
Let γ denote a curve of osculating circles to a curve C in E2 or E3.
Then the 1-form 4√⟨d2γdt2,d2γdt2⟩dt is independent of the parameter t.
Let ρ be a parameter so that the dρ=4√⟨d2γdt2,d2γdt2⟩dt. In other words, the parameter ρ can be characterized by

⟨\tiny∘∘γ,\tiny∘∘γ⟩=1,

(2.14)

where \tiny∘∘γ=d2γ/dρ2.
It can be uniquely determined up to ρ↦±ρ+c for some constant c.

This parameter is called the conformal arc-length of C.
Our assumption that the curve C is vertex-free guarantees that ρ serves as a non-singular parameter of C.

Let γ⊂Λ3 be an osculating circle to a curve C in E2 at a point m.
Then they form a curve in de Sitter space Λ3.
We express the derivative by the conformal arc-length ρ by putting \tiny∘σ above.

3.1 The moving frame and Frenet formula

Our moving frames consist of two space-like vectors and two light-like vectors, instead of three space-like vectors and a time-like vector, since we take a light-like vector in span(m) as our first frame, which we denote by n.
Another light-like vector, which comes last in our frames, is chosen so that ⟨n,n∗⟩=−1. The middle two space-like vectors of our frames are taken from an orthonormal basis of (span(n,n∗))⊥.
The moving frames of this form are called isotropic orthonormal frames.

Let us choose the first vector n of our moving frames in span(m) so that ∥\tiny∘n∥=1.
Then we can take n=\tiny∘γ.
Our second vector v1 is \tiny∘n.
Then the x-coordinate of the normal form can be given by ⟨m(ρ),v1⟩.
We choose our third vector v2 in (span(m))⊥ so that ⟨vi,vj⟩=δij and that the y coordinate of the normal form can be given by ⟨m(ρ),v2⟩.
The x-axis of the normal form should be the osculating circle.
Therefore, we can take v2=γ.
Our last vector n∗ is a light-like vector in (span(v1,v2))⊥ that satisfies ⟨n,n∗⟩=−1.

Then ⟨\tiny∘∘∘γ,\tiny∘∘∘γ⟩=−2Q2.
Lemma 2.1 and proposition 2.2 imply that the ⟨\largediγdρi,djγdρj⟩ for small i,j are given by table 1.
It implies that our last frame is given by n∗=−Q2\tiny∘γ+\tiny∘∘∘γ=−Q2n+\tiny∘∘n.

3.2 Normal form

The normal form of a planar curve can be obtained from that of a space curve (see subsection 4.3) by putting T=0 and Q=Q2 and forgetting the z-coordinate.
If we only look for the normal form of a plane curve, the computation is simpler.
Thanks to table 1, we do not have to rewrite n(i)=γ(i+1) in terms of the frames.
The direction vectors of x- and y- axes at ρ=0 are given by \tiny∘∘γ(0) and γ(0) respectively.

4.1 Osculating spheres and conformal torsion

Let m be a point on a curve C in E3.
Let γ⊂˜G+2,5 be a curve of osculating circles, and σ⊂Λ4 a curve of osculating spheres.
Since an osculating sphere intersects an infinitesimally close osculating sphere in an osculating circle, σ is a space-like curve in Λ4.
Let ρ be the conformal arc-length and l the arc-length parameter of the curve σ.
We express the derivatives by l and ρ by putting above \Huge.σ and \tiny∘σ respectively.
Put

T=dldρ.

(4.1)

As dl measures the infinitesimal angle variation, T measures how an osculating sphere rotates around an osculating circle with respect to the conformal arc-length.
For simplicity’s sake, let us assume T>0 in what follows.
It is proved in [RS] that T coincides with the conformal torsion up to sign.
As is pointed out in [Sh], the conformal torsion can be determined up to sign.
We remark that the conformal torsion is identically equal to 0 if and only if C is a planar or a spherical curve.

Lemma 4.1

We have ⟨\Huge.\kern-3.0pt.σ,\Huge.%
\kern-3.0pt.σ⟩=1.

Proof:
Since ⟨σ,σ⟩=1 and ⟨\Huge.σ,\Huge.σ⟩=1, we have ⟨σ,\Huge.σ⟩=0 and ⟨σ,\Huge.\kern-3.0pt.σ⟩=−1.
On the other hand, since γ=σ∧\Huge.σ we have \Huge.γ=σ∧\Huge.\kern-3.0%
pt.σ.
Therefore, the formula (2.2) implies

Proposition 4.2

Proof:
The first equality comes from proposition 2.2, the fact that dρ=4√⟨d2γdt2,d2γdt2⟩dt for any parameter t ([LO2]).

Since ⟨σ,σ⟩=⟨\Huge.σ,\Huge.σ⟩=⟨\Huge.\kern-3.0pt.σ,\Huge.\kern-3.0pt.σ⟩=1, we have
⟨σ,\Huge.σ⟩=⟨%
\Huge.σ,\Huge.\kern-3.0pt.σ⟩=⟨\Huge.\kern-3.0pt.σ,\Huge.\kern-3.0%
pt.\kern-3.0pt.σ⟩=0 and hence
⟨σ,\Huge.\kern-3.0pt.σ⟩=⟨\Huge.σ,\Huge.\kern-3.0pt.\kern-3.0pt%
.σ⟩=−1 and ⟨σ,\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩=0.
Therefore the formula (2.2) implies

As γ=σ∧\Huge.σ and therefore \Huge.\kern-3.0pt.γ=σ∧%
\Huge.\kern-3.0pt.\kern-3.0pt.σ+\Huge.σ∧\Huge.\kern-3.0pt.σ,
we have ⟨\Huge.\kern-3.0pt.γ,\Huge.%
\kern-3.0pt.γ⟩=⟨\Huge.\kern-3.0pt.\kern-%
3.0pt.σ,\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩−1, which completes the proof.
□

Lemma 4.3

Under our assumption, i.e. if the curve C is vertex-free and dldρ never vanishes, the five vectors σ,\Huge.σ,\Huge.\kern-3.0pt.σ,\Huge.\kern-3.0pt.\kern-3.0pt.σ, and σ(4) are linearly independent.

Proof:
Suppose a0σ+a1\Huge.σ+a2\Huge.%
\kern-3.0pt.σ+a3\Huge.\kern-3.0pt.\kern-3.0pt.σ+a4σ(4)=\boldmath0 for some a0,⋯a4∈R.
Put G3=⟨\Huge.\kern-3.0pt.\kern-3.0pt.σ,\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩ and G4=⟨σ(4),σ(4)⟩.
Then, by taking pseudo inner product with σ,⋯,σ(4), we have

The determinant of the coefficient matrix is equal to (1−G3)3=−T−12≠0. □

4.2 Moving frame in R51 and Frenet formula

Proposition 4.4

([Y])
A point m on the curve C can be expressed in terms of the osculating spheres as

m=c(σ+\Huge.\kern-3.0pt.σ)(c∈R∖{0}).

(4.2)

Proof:
Since ⟨σ,\Huge.\kern-3.0pt.σ⟩=−1 and ⟨\Huge.\kern-3.0pt.σ,\Huge.%
\kern-3.0pt.σ⟩=1 by lemma 4.1 and its proof, σ+\Huge.\kern-3.0pt.σ is a light-like vector.

On the other hand, as σ can be expressed as σ=φm×\Huge.m×\Huge%
.\kern-3.0pt.m×\Huge.\kern-3.0pt.\kern-3.0pt.m for some function φ, \Huge.\kern-3.0pt.σ can be expressed as a linear combination of vectors of the form m×\Huge.m×m(i)×m(j).
Therefore, ⟨m,σ+\Huge.\kern-3.0pt.σ⟩=0, which completes the proof, as (span(m))⊥ is tangent to the light cone in span(m).
□

We choose the conformal arc-length, not l, for the parameter of the Frenet formula, as is the case in most preceding studies.
Let us choose the first vector n of our moving frames in span(m) so that ∥\tiny∘n∥=1.
Then we can take n=T(σ+\Huge.\kern-3.0pt.σ).
Our second vector v1 is \tiny∘n.
Then the x-coordinate of the normal form can be given by ⟨m(ρ),v1⟩.
We choose our third and fourth vectors v2, and v3 in (span(m))⊥ so that ⟨vi,vj⟩=δij and that the y, and z coordinates of the normal form can be given by ⟨m(ρ),v2⟩, and ⟨m(ρ),v3⟩ respectively.
The xy-plane of the normal form should be the osculating sphere.
Therefore we can take v3=−σ. The choice of the frame is important to get the same normal form as in [CSW]. We put − here so that the Frenet matrix and the normal form fit with those in [Su] and [CSW] respectively.
A sphere which corresponds to v2 should intersect the osculating sphere orthogonally in the osculating circle.
Therefore we can take v2=\Huge.σ.
Our last vector n∗ is a light-like vector in (span(v1,v2,v3))⊥ that satisfies ⟨n,n∗⟩=−1.

also serve as moving frames of the curve σ⊂Λ4 of osculating spheres.
Note that n is not necessarily equal to a point m in the curve C⊂E3.
In order to have ∥\tiny∘n∥=1, we enlarge or shrink the point vector of a curve in the light cone keeping the parameter to be the conformal arc-length.

Proof:
The formula (4.1) implies \tiny∘u=T\Huge.u for any u.
The first, third, and fourth rows of the matrix follow from the definition of the frames.
The other two rows can be obtained by derivating the scalar products of the frames, i.e. ⟨n,v1⟩ etc.
Remark that “n- (or n∗-)coordinate” of a vector u can be given by −⟨u,n∗⟩ (or −⟨u,n⟩ respectively).
□

Since \tiny∘∘n=\tiny∘v1=Qn+n∗ we have ⟨\tiny∘∘n,\tiny∘∘n⟩=⟨Qn+n∗,Qn+n∗⟩=−2Q.
Therefore,

Q=−12⟨\tiny∘∘n,\tiny∘∘n⟩.

(4.5)

This quantity Q is called the conformal curvature (denoted by λ1 in [Su], whereas the conformal torsion T is denoted by ±λ2 in [Su]).

As \tiny∘∘n=Qn+n∗, our last frame is given by n∗=−Qn+\tiny∘∘n as in the case of planar curve.
Thus our isotropic orthonormal moving frames are

n∈span(m),v1=\tiny∘n,v2=\tiny∘σ∥\tiny∘σ∥,v3=−σ,n∗=−Qn+\tiny∘∘n.

Theorem 4.6

Let γ⊂˜G+2,5 be a curve of osculating circles and σ⊂Λ4 a curve of osculating spheres.
Then the conformal curvature Q and the conformal torsion T satisfy

T

=

√⟨\tiny∘σ,\tiny∘σ⟩,

(4.6)

Q

=

−12⟨\tiny∘∘∘γ,\tiny∘∘∘γ⟩+3⟨\tiny∘σ,\tiny∘σ⟩.

(4.7)

Proof:
The first equation is trivial from the definition (4.1) of T as ⟨\tiny∘σ,\tiny∘% σ⟩=⟨T\Huge.σ,T\Huge.σ⟩=T2.